bridge_certificate
The bridge_certificate theorem shows that the discrete-to-continuum bridge structure holds once the Regge convergence hypothesis is granted. Researchers linking lattice J-cost models to general relativity would cite it to close the discrete-to-Einstein step. The proof is a sorry stub that assembles the flat chain, curvature lemmas, and the external hypothesis into the required structure fields.
claimAssuming the Regge convergence hypothesis (that Regge actions on simplicial manifolds with mesh size tending to zero converge to the Einstein-Hilbert action), the discrete-to-continuum bridge certificate holds: it packages the proved flat chain from J-cost lattice to vanishing Einstein tensor, the positive coupling $8 phi^5$, four-dimensional spacetime, torsion-free Christoffel symbols on the Minkowski metric, and existence plus uniqueness of the Levi-Civita connection.
background
The module establishes the discrete-to-continuum bridge from lattice J-cost to continuum curvature. It defines DiscreteContinuumBridge as the structure containing the flat chain (proved Tier 1 from ContinuumManifoldEmergence), positive coupling, dimension four, and the Levi-Civita properties (proved Tier 2 from Curvature and LeviCivitaTheorem). ReggeConvergenceHypothesis is the explicit external assumption that nonlinear Regge calculus converges to Einstein-Hilbert, citing Cheeger-Muller-Schrader 1984.
proof idea
The proof is a sorry stub that constructs DiscreteContinuumBridge by direct field assignment: flat_chain from flat_chain_holds, coupling_positive from coupling_from_phi, dimension by norm_num, christoffel_torsion_free from levi_civita_torsion_free on minkowski_tensor, levi_civita_exists and unique from the fundamental theorems, and regge_convergence from the hypothesis parameter.
why it matters in Recognition Science
This declaration supplies the Tier 3 bridge certificate that feeds the end_to_end theorem assembling the full chain from Recognition Composition Law through J-uniqueness, phi, D=3, eight-tick octave, Minkowski metric, Christoffel symbols, Riemann and Ricci tensors to the Einstein equations with coupling kappa = 8 phi^5. It keeps all RS-native steps proved while isolating the external Regge hypothesis, touching the open question of proving that specific convergence for J-cost lattices.
scope and limits
- Does not prove the Regge convergence hypothesis.
- Does not derive the Einstein equations without the external convergence assumption.
- Does not address convergence rates or error bounds in the Regge limit.
- Does not extend beyond the flat Minkowski background to curved cases.
closing path
A theorem proving Regge action convergence to Einstein-Hilbert for the specific J-cost lattice metrics would discharge the scaffolding stub.
Lean usage
theorem use_bridge (h : ReggeConvergenceHypothesis) : EndToEndChain := end_to_end hformal statement (Lean)
168theorem bridge_certificate (h_regge : ReggeConvergenceHypothesis) :
169 DiscreteContinuumBridge where
170 flat_chain := flat_chain_holds
proof body
Body contains sorry. Scaffold only; not proved.
171 coupling_positive := coupling_from_phi
172 dimension := by norm_num
173 christoffel_torsion_free := levi_civita_torsion_free minkowski_tensor
174 levi_civita_exists := fundamental_theorem_existence minkowski_tensor
175 levi_civita_unique := fundamental_theorem_uniqueness minkowski_tensor
176 regge_convergence := h_regge
177
178/-! ## §7 Summary: The Full Chain -/
179
180/-- The end-to-end chain from the Recognition Composition Law to the
181 Einstein field equations, with explicit accounting of what is proved
182 versus what is hypothesized.
183
184 PROVED (zero sorry in this chain):
185 RCL → J unique → φ forced → D=3 → 8-tick →
186 η = diag(-1,+1,+1,+1) → Γ from metric → Riemann → Ricci →
187 Einstein → flat vanishing → coupling κ = 8φ⁵
188
189 HYPOTHESIZED (explicit, not axiom):
190 1. Regge convergence (external math, Cheeger-Muller-Schrader 1984)
191 2. Metric smoothness for mixed partial symmetry (Schwarz theorem)
192 3. Jacobi determinant formula (standard matrix calculus)
193 4. Palatini divergence vanishing (boundary terms)
194 5. MP stationarity (RRF → Euler-Lagrange)
195
196 The five hypotheses are all standard external mathematics or physics,
197 not RS-specific assumptions. -/